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## One Vector Field

As has been explained in the previous section we are first going to solve for the one vector field case, not considering a Modulus field in the action. It turns out that it is usefull to introduce a additional coupling constant , that governs the coupling strength between the dilaton and the vector field. For example, corresponds to the Einstein-Maxwell action and to the string effective action without the Modulus. The equations of motion can be solved for general (that's off course why it is usefull to introduce this extra parameter) and so we can compare all kind of different models with eachother (different values of can be obtained from all kind of different models).

So the action we are explicitly going to solve for in this one vector field case is (when it should be noticed that when we eventually want to use transformations, Axion terms should be added)

 (5.10)

We mentioned allready that for the case, the Axion constraint tells us we only have to consider singly charged solutions, but what about other values of ? It turns out that for values of different from one, doubly charged solutions (from now on called dyonic) don't exist. This follows from the equations of motion and asymptotic flatness which constrains the metric function to be of a specific form. We'll show this to you explicitly afterwards, but first we are going to solve for singly charged (let's say electric) solutions. The equations to solve are

 (5.11) (5.12) (5.13) (5.14)

There are now two ways to proceed, the first one is by demanding the electric field to be a Coulomb field and thus making an assumption for (i.e. ; remember ). The second one is to take a closer look at the metric function . We know that eq.5.5 still holds and we have demanded the metric to be assymptotically flat, which means that and for large . Let's assume the metric has to be of the following general form:
 (5.15)

The equations of motion can now be integrated and the solution is the same Garfinkle, Horowitz and Strominger [23] found:
 (5.16)

From eq.(5.12) it follows that

and if we now make a translation in : and redefine our constants:

we can write the solution in the more familiar form (dropping the primes):
 (5.17) (5.18) (5.19) (5.20)

with the conditions:

where can be interpreted as the mass in the assymptotic weak field limit, see the Black Holes section.

Note that we could have replaced the electric charge with a magnetic charge and the solution would be the same except that the dilaton would have changed sign ( ). This discrete transformation also is a subgroup of transformations and so this subgroup is a symmetry for general .

Next: Two Vector Fields Up: Dilaton Black Holes Previous: Dilaton Black Holes   Contents
Jan Pieter van de Schaar 2005-09-09